Fishnet model with order statistics for tail probability of failure of nacreous biomimetic materials with softening interlaminar links

The staggered (or imbricated) lamellar “brick-and-mortar” nanostructure of nacre endows nacre with strength and fracture toughness values exceeding by an order of magnitude those of the constituents, and inspires the advent of new robust biomimetic materials. While many deterministic studies clarified these advantageous features in the mean sense, a closed-form statistical model is indispensable for determining the tail probability of failure in the range of 1 in a million, which is what is demanded for most engineering applications. In the authors' preceding study, the so-called 'fishnet' statistics, exemplified by a diagonally pulled fishnet, was conceived to describe the probability distribution. The fishnet links, representing interlaminar bonds, were considered to be elastic perfectly-brittle. However, the links may often be quasibrittle or almost ductile, exhibiting gradual postpeak softening in their stress-strain relation. This paper extends the fishnet statistics to links with post-peak softening slope of arbitrary steepness. Probabilistic analysis is enabled by assuming the postpeak softening of a link to occur as a series of finite drops of stress and stiffness. The maximum load of the structure is approximated by the strength of the kth weakest link (k ≥ 1), and the distribution of structure strength is expressed as a weighted sum of the distributions of order statistics. The analytically obtained probabilities are compared and verified by histograms of strength data obtained by millions of Monte Carlo simulations for each of many nacreous bodies with different link softening steepness and with various overall shapes.